Not sure what you mean by a "payment request". What you should have sent him is an invoice. In general, a business will not make a payment unless there is an invoice to back it up.
Also, the invoice should state the terms of payment, that is, how many days or weeks you will allow for the payment to reach you.
On that point, one week is too soon to chase payment, unless that period was specifically agreed by the client at the outset. Most companies process payments on a monthly cycle, which means it could take anywhere from 30 to 60 days for the money to be sent to you. You've then got to allow for the time it takes the bank to process it, or the postal service to deliver it.
Finally, sending a second invoice is not the way to chase payment if the first one wasn't paid. Better to send a friendly reminder. If that doesn't work, send a more insistent reminder (but always a polite one). Sending a second invoice will just confuse their payment system, and therefore risk a further delay (not to mention making you look unprofessional).